Not much to say there. So far, the 12-month moving average shows no sign of any recovery. Arctic sea ice extent remains far below the 1979 start point or even where it was before 2005. However, the claim is that the ice gain since September 2012 set a record. Normally, Arctic ice extent reaches the yearly minimum in September at the end of summer, with a maximum the following March at the end of winter. The ice gain is the difference between those months.
March extent has been declining linearly by an average of -36,581 km2 per year whereas the decline in September has been by an average of -87,112 km2 and accelerating by -7,786 km2 per year. As a result of the accelerating losses in September, the yearly difference between March and September has been growing at an accelerating rate.
The second claim that Arctic sea ice is 60% higher in August 2013 than it was in August 2012 is partially true—the ice was higher in August 2013. But it wasn't by 60%. Average ice extent in August 2013 was 6.05 million km2. Ice extent in August 2012 was 4.72 million km2. That is a difference of 1.33 million km2, which means that the ice in August 2013 was 28.2% higher than it was in August 2012 (1.33/4.72 * 100). As for the claim that August extent is the highest in years, here's a graph of August sea ice extent since 1979.
Finally, sea ice extent is not the full story in the Arctic. A larger but less noticeable change is that the sea ice has become much thinner.
The ice has lost nearly 12,000 km3 of volume since 1979. Even if ice extent does grow, all it means is that the surface is covered with thinner, less stable ice than it used to be. And thinner ice is more susceptible to melting when conditions are right.
Lastly, there's a little concept in statistics called "regression toward the mean". In simple terms, data tends to fall around the average, with extreme outliers followed by data points that are closer to the average. How is that relevant to Arctic sea ice extent? September 2012 was quite a bit below the trend line. It's an outlier. Therefore, we'd expect the monthly average for September 2013 to be closer to the average. Extrapolating from the trend line gives an expected September 2013 extent of 4.15 million km2. That represents a 16% increase over the average in September 2012 (3.58 million km2) even if the overall trend remains the exact same. Figuring in the 95% confidence interval for the trend (4.66 million km2) shows that the average September 2013 sea ice could show an increase of 30% even if the trend remains the exact same. This doesn't even consider the standard deviation of the data around the trend (± 1.32 million square kilometers), which means that 2013 September ice extent could be as high as 53% higher (up to 5.47 million square kilometers) than in 2012, even if the current trend, standard error, and standard deviation are unchanged.
The main message from the Arctic is simple: The ice continues to melt, regardless of what tabloid articles and deniers say.